Since in the large deviation regime the average and the most probable
value coincide as a first approximation, this means that the time
average of the observable in the effective ensemble will fluctuate
around a typical value given by the derivative of the scaled cumulant
generating function evaluated in \(k^{*}\) . This is exactly the range
of fluctuations that are needed in order to compute correctly the
large deviation function in a neighborhood of \(k^{*}\)  [1]}.
Effectively tilting the trajectory probability density shifts the
center of the convergence region of the estimator around \(k=k^{*}\) .
It is therefore possible to perform a few experiments with different
values of \(k^{*}\) , compute the estimate of \(\lambda (k)\)  for the
neighborhoods of the values of \(k^{*},\)  and then join the estimates
to reconstruct \(\lambda (k)\)  piecewise. This allows to explore the
tails of the scaled cumulant generating function (hence of the rate
function) with a huge gain in terms of computational cost with respect
to direct estimation methods.
